INSTRUCTORS Carleen Eaton Grant Fraser

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Dr. Carleen Eaton

Dr. Carleen Eaton

Arithmetic Series

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Table of Contents

Section 1: Equations and Inequalities
Expressions and Formulas

22m 23s

Intro
0:00
Order of Operations
0:19
Variable
0:27
Algebraic Expression
0:46
Term
0:57
Example: Algebraic Expression
1:25
Evaluate Inside Grouping Symbols
1:55
Evaluate Powers
2:30
Multiply/Divide Left to Right
2:55
Add/Subtract Left to Right
3:35
Monomials
4:40
Examples of Monomials
4:52
Constant
5:27
Coefficient
5:46
Degree
6:25
Power
7:15
Polynomials
8:02
Examples of Polynomials
8:24
Binomials, Trinomials, Monomials
8:53
Term
9:21
Like Terms
10:02
Formulas
11:00
Example: Pythagorean Theorem
11:15
Example 1: Evaluate the Algebraic Expression
11:50
Example 2: Evaluate the Algebraic Expression
14:38
Example 3: Area of a Triangle
19:11
Example 4: Fahrenheit to Celsius
20:41
Properties of Real Numbers

20m 15s

Intro
0:00
Real Numbers
0:07
Number Line
0:15
Rational Numbers
0:46
Irrational Numbers
2:24
Venn Diagram of Real Numbers
4:03
Irrational Numbers
5:00
Rational Numbers
5:19
Real Number System
5:27
Natural Numbers
5:32
Whole Numbers
5:53
Integers
6:19
Fractions
6:46
Properties of Real Numbers
7:15
Commutative Property
7:34
Associative Property
8:07
Identity Property
9:04
Inverse Property
9:53
Distributive Property
11:03
Example 1: What Set of Numbers?
12:21
Example 2: What Properties Are Used?
13:56
Example 3: Multiplicative Inverse
16:00
Example 4: Simplify Using Properties
17:18
Solving Equations

19m 10s

Intro
0:00
Translations
0:06
Verbal Expressions and Algebraic Expressions
0:13
Example: Sum of Two Numbers
0:19
Example: Square of a Number
1:33
Properties of Equality
3:20
Reflexive Property
3:30
Symmetric Property
3:42
Transitive Property
4:01
Addition Property
5:01
Subtraction Property
5:37
Multiplication Property
6:02
Division Property
6:30
Solving Equations
6:58
Example: Using Properties
7:18
Solving for a Variable
8:25
Example: Solve for Z
8:34
Example 1: Write Algebraic Expression
10:15
Example 2: Write Verbal Expression
11:31
Example 3: Solve the Equation
14:05
Example 4: Simplify Using Properties
17:26
Solving Absolute Value Equations

17m 31s

Intro
0:00
Absolute Value Expressions
0:09
Distance from Zero
0:18
Example: Absolute Value Expression
0:24
Absolute Value Equations
1:50
Example: Absolute Value Equation
2:00
Example: Isolate Expression
3:13
No Solution
3:46
Empty Set
3:58
Example: No Solution
4:12
Number of Solutions
4:46
Check Each Solution
4:57
Example: Two Solutions
5:05
Example: No Solution
6:18
Example: One Solution
6:28
Example 1: Evaluate for X
7:16
Example 2: Write Verbal Expression
9:08
Example 3: Solve the Equation
12:18
Example 4: Simplify Using Properties
13:36
Solving Inequalities

17m 14s

Intro
0:00
Properties of Inequalities
0:08
Addition Property
0:17
Example: Using Numbers
0:30
Subtraction Property
1:03
Example: Using Numbers
1:19
Multiplication Properties
1:44
C>0 (Positive Number)
1:50
Example: Using Numbers
2:05
C<0 (Negative Number)
2:40
Example: Using Numbers
3:10
Division Properties
4:11
C>0 (Positive Number)
4:15
Example: Using Numbers
4:27
C<0 (Negative Number)
5:21
Example: Using Numbers
5:32
Describing the Solution Set
6:10
Example: Set Builder Notation
6:26
Example: Graph (Closed Circle)
7:08
Example: Graph (Open Circle)
7:30
Example 1: Solve the Inequality
7:58
Example 2: Solve the Inequality
9:06
Example 3: Solve the Inequality
10:10
Example 4: Solve the Inequality
13:12
Solving Compound and Absolute Value Inequalities

25m

Intro
0:00
Compound Inequalities
0:08
And and Or
0:13
Example: And
0:22
Example: Or
1:12
And Inequality
1:41
Intersection
1:49
Example: Numbers
2:08
Example: Inequality
2:43
Or Inequality
4:35
Example: Union
4:45
Example: Inequality
5:53
Absolute Value Inequalities
7:19
Definition of Absolute Value
7:33
Examples: Compound Inequalities
8:30
Example: Complex Inequality
12:21
Example 1: Solve the Inequality
12:54
Example 2: Solve the Inequality
17:21
Example 3: Solve the Inequality
18:54
Example 4: Solve the Inequality
22:15
Section 2: Linear Relations and Functions
Relations and Functions

32m 5s

Intro
0:00
Coordinate Plane
0:20
X-Coordinate and Y-Coordinate
0:30
Example: Coordinate Pairs
0:37
Quadrants
1:20
Relations
2:14
Domain and Range
2:19
Set of Ordered Pairs
2:29
As a Table
2:51
Functions
4:21
One Element in Range
4:32
Example: Mapping
4:43
Example: Table and Map
6:26
One-to-One Functions
8:01
Example: One-to-One
8:22
Example: Not One-to-One
9:18
Graphs of Relations
11:01
Discrete and Continuous
11:12
Example: Discrete
11:22
Example: Continous
12:30
Vertical Line Test
14:09
Example: S Curve
14:29
Example: Function
16:15
Equations, Relations, and Functions
17:03
Independent Variable and Dependent Variable
17:16
Function Notation
19:11
Example: Function Notation
19:23
Example 1: Domain and Range
20:51
Example 2: Discrete or Continous
23:03
Example 3: Discrete or Continous
25:53
Example 4: Function Notation
30:05
Linear Equations

14m 46s

Intro
0:00
Linear Equations and Functions
0:07
Linear Equation
0:19
Example: Linear Equation
0:29
Example: Linear Function
1:07
Standard Form
2:02
Integer Constants with No Common Factor
2:08
Example: Standard Form
2:27
Graphing with Intercepts
4:05
X-Intercept and Y-Intercept
4:12
Example: Intercepts
4:26
Example: Graphing
5:14
Example 1: Linear Function
7:53
Example 2: Linear Function
9:10
Example 3: Standard Form
10:04
Example 4: Graph with Intercepts
12:25
Slope

23m 7s

Intro
0:00
Definition of Slope
0:07
Change in Y / Change in X
0:26
Example: Slope of Graph
0:37
Interpretation of Slope
3:07
Horizontal Line (0 Slope)
3:13
Vertical Line (Undefined Slope)
4:52
Rises to Right (Positive Slope)
6:36
Falls to Right (Negative Slope)
6:53
Parallel Lines
7:18
Example: Not Vertical
7:30
Example: Vertical
7:58
Perpendicular Lines
8:31
Example: Perpendicular
8:42
Example 1: Slope of Line
10:32
Example 2: Graph Line
11:45
Example 3: Parallel to Graph
13:37
Example 4: Perpendicular to Graph
17:57
Writing Linear Functions

23m 5s

Intro
0:00
Slope Intercept Form
0:11
m and b
0:28
Example: Graph Using Slope Intercept
0:43
Point Slope Form
2:41
Relation to Slope Formula
3:03
Example: Point Slope Form
4:36
Parallel and Perpendicular Lines
6:28
Review of Parallel and Perpendicular Lines
6:31
Example: Parallel
7:50
Example: Perpendicular
9:58
Example 1: Slope Intercept Form
11:07
Example 2: Slope Intercept Form
13:07
Example 3: Parallel
15:49
Example 4: Perpendicular
18:42
Special Functions

31m 5s

Intro
0:00
Step Functions
0:07
Example: Apple Prices
0:30
Absolute Value Function
4:55
Example: Absolute Value
5:05
Piecewise Functions
9:08
Example: Piecewise
9:27
Example 1: Absolute Value Function
14:00
Example 2: Absolute Value Function
20:39
Example 3: Piecewise Function
22:26
Example 4: Step Function
25:25
Graphing Inequalities

21m 42s

Intro
0:00
Graphing Linear Inequalities
0:07
Shaded Region
0:19
Using Test Points
0:32
Graph Corresponding Linear Function
0:46
Dashed or Solid Lines
0:59
Use Test Point
1:21
Example: Linear Inequality
1:58
Graphing Absolute Value Inequalities
4:50
Graph Corresponding Equations
4:59
Use Test Point
5:20
Example: Absolute Value Inequality
5:38
Example 1: Linear Inequality
9:17
Example 2: Linear Inequality
11:56
Example 3: Linear Inequality
14:29
Example 4: Absolute Value Inequality
17:06
Section 3: Systems of Equations and Inequalities
Solving Systems of Equations by Graphing

17m 13s

Intro
0:00
Systems of Equations
0:09
Example: Two Equations
0:24
Solving by Graphing
0:53
Point of Intersection
1:09
Types of Systems
2:29
Independent (Single Solution)
2:34
Dependent (Infinite Solutions)
3:05
Inconsistent (No Solution)
4:23
Example 1: Solve by Graphing
5:20
Example 2: Solve by Graphing
9:10
Example 3: Solve by Graphing
12:27
Example 4: Solve by Graphing
14:54
Solving Systems of Equations Algebraically

23m 53s

Intro
0:00
Solving by Substitution
0:08
Example: System of Equations
0:36
Solving by Multiplication
7:22
Extra Step of Multiplying
7:38
Example: System of Equations
8:00
Inconsistent and Dependent Systems
11:14
Variables Drop Out
11:48
Inconsistent System (Never True)
12:01
Constant Equals Constant
12:53
Dependent System (Always True)
13:11
Example 1: Solve Algebraically
13:58
Example 2: Solve Algebraically
15:52
Example 3: Solve Algebraically
17:54
Example 4: Solve Algebraically
21:40
Solving Systems of Inequalities By Graphing

27m 12s

Intro
0:00
Solving by Graphing
0:08
Graph Each Inequality
0:25
Overlap
0:35
Corresponding Linear Equations
1:03
Test Point
1:23
Example: System of Inequalities
1:51
No Solution
7:06
Empty Set
7:26
Example: No Solution
7:34
Example 1: Solve by Graphing
10:27
Example 2: Solve by Graphing
13:30
Example 3: Solve by Graphing
17:19
Example 4: Solve by Graphing
23:23
Solving Systems of Equations in Three Variables

28m 53s

Intro
0:00
Solving Systems in Three Variables
0:17
Triple of Values
0:31
Example: Three Variables
0:56
Number of Solutions
5:55
One Solution
6:08
No Solution
6:24
Infinite Solutions
7:06
Example 1: Solve 3 Variables
7:59
Example 2: Solve 3 Variables
13:50
Example 3: Solve 3 Variables
19:54
Example 4: Solve 3 Variables
25:50
Section 4: Matrices
Basic Matrix Concepts

11m 34s

Intro
0:00
What is a Matrix
0:26
Brackets
0:46
Designation
1:21
Element
1:47
Matrix Equations
1:59
Dimensions
2:27
Rows (m) and Columns (n)
2:37
Examples: Dimensions
2:43
Special Matrices
4:22
Row Matrix
4:32
Column Matrix
5:00
Zero Matrix
6:00
Equal Matrices
6:30
Example: Corresponding Elements
6:36
Example 1: Matrix Dimension
8:12
Example 2: Matrix Dimension
9:03
Example 3: Zero Matrix
9:38
Example 4: Row and Column Matrix
10:26
Matrix Operations

21m 36s

Intro
0:00
Matrix Addition
0:18
Same Dimensions
0:25
Example: Adding Matrices
1:04
Matrix Subtraction
3:42
Same Dimensions
3:48
Example: Subtracting Matrices
4:04
Scalar Multiplication
6:08
Scalar Constant
6:24
Example: Multiplying Matrices
6:32
Properties of Matrix Operations
8:23
Commutative Property
8:41
Associative Property
9:08
Distributive Property
9:44
Example 1: Matrix Addition
10:24
Example 2: Matrix Subtraction
11:58
Example 3: Scalar Multiplication
14:23
Example 4: Matrix Properties
16:09
Matrix Multiplication

29m 36s

Intro
0:00
Dimension Requirement
0:17
n = p
0:24
Resulting Product Matrix (m x q)
1:21
Example: Multiplication
1:54
Matrix Multiplication
3:38
Example: Matrix Multiplication
4:07
Properties of Matrix Multiplication
10:46
Associative Property
11:00
Associative Property (Scalar)
11:28
Distributive Property
12:06
Distributive Property (Scalar)
12:30
Example 1: Possible Matrices
13:31
Example 2: Multiplying Matrices
17:08
Example 3: Multiplying Matrices
20:41
Example 4: Matrix Properties
24:41
Determinants

33m 13s

Intro
0:00
What is a Determinant
0:13
Square Matrices
0:23
Vertical Bars
0:41
Determinant of a 2x2 Matrix
1:21
Second Order Determinant
1:37
Formula
1:45
Example: 2x2 Determinant
1:58
Determinant of a 3x3 Matrix
2:50
Expansion by Minors
3:08
Third Order Determinant
3:19
Expanding Row One
4:06
Example: 3x3 Determinant
6:40
Diagonal Method for 3x3 Matrices
13:24
Example: Diagonal Method
13:36
Example 1: Determinant of 2x2
18:59
Example 2: Determinant of 3x3
20:03
Example 3: Determinant of 3x3
25:35
Example 4: Determinant of 3x3
29:22
Cramer's Rule

28m 25s

Intro
0:00
System of Two Equations in Two Variables
0:16
One Variable
0:50
Determinant of Denominator
1:14
Determinants of Numerators
2:23
Example: System of Equations
3:34
System of Three Equations in Three Variables
7:06
Determinant of Denominator
7:17
Determinants of Numerators
7:52
Example 1: Two Equations
8:57
Example 2: Two Equations
13:21
Example 3: Three Equations
17:11
Example 4: Three Equations
23:43
Identity and Inverse Matrices

22m 25s

Intro
0:00
Identity Matrix
0:13
Example: 2x2 Identity Matrix
0:30
Example: 4x4 Identity Matrix
0:50
Properties of Identity Matrices
1:24
Example: Multiplying Identity Matrix
2:52
Matrix Inverses
5:30
Writing Matrix Inverse
6:07
Inverse of a 2x2 Matrix
6:39
Example: 2x2 Matrix
7:31
Example 1: Inverse Matrix
10:18
Example 2: Find the Inverse Matrix
13:04
Example 3: Find the Inverse Matrix
17:53
Example 4: Find the Inverse Matrix
20:44
Solving Systems of Equations Using Matrices

22m 32s

Intro
0:00
Matrix Equations
0:11
Example: System of Equations
0:21
Solving Systems of Equations
4:01
Isolate x
4:16
Example: Using Numbers
5:10
Multiplicative Inverse
5:54
Example 1: Write as Matrix Equation
7:18
Example 2: Use Matrix Equations
9:12
Example 3: Use Matrix Equations
15:06
Example 4: Use Matrix Equations
19:35
Section 5: Quadratic Functions and Inequalities
Graphing Quadratic Functions

31m 48s

Intro
0:00
Quadratic Functions
0:12
A is Zero
0:27
Example: Parabola
0:45
Properties of Parabolas
2:08
Axis of Symmetry
2:11
Vertex
2:32
Example: Parabola
2:48
Minimum and Maximum Values
9:02
Positive or Negative
9:28
Upward or Downward
9:58
Example: Minimum
10:31
Example: Maximum
11:16
Example 1: Axis of Symmetry, Vertex, Graph
12:41
Example 2: Axis of Symmetry, Vertex, Graph
17:25
Example 3: Minimum or Maximum
21:47
Example 4: Minimum or Maximum
27:09
Solving Quadratic Equations by Graphing

27m 3s

Intro
0:00
Quadratic Equations
0:16
Standard Form
0:18
Example: Quadratic Equation
0:47
Solving by Graphing
1:41
Roots (x-Intercepts)
1:48
Example: Number of Solutions
2:12
Estimating Solutions
9:23
Example: Integer Solutions
9:30
Example: Estimating
9:53
Example 1: Solve by Graphing
10:52
Example 2: Solve by Graphing
15:10
Example 1: Solve by Graphing
17:50
Example 1: Solve by Graphing
20:54
Solving Quadratic Equations by Factoring

19m 53s

Intro
0:00
Factoring Techniques
0:15
Greatest Common Factor (GCF)
0:37
Difference of Two Squares
1:48
Perfect Square Trinomials
2:30
General Trinomials
3:09
Zero Product Rule
5:22
Example: Zero Product
5:53
Example 1: Solve by Factoring
7:46
Example 1: Solve by Factoring
9:48
Example 1: Solve by Factoring
12:34
Example 1: Solve by Factoring
15:28
Imaginary and Complex Numbers

35m 45s

Intro
0:00
Properties of Square Roots
0:10
Product Property
0:26
Example: Product Property
0:56
Quotient Property
2:17
Example: Quotient Property
2:35
Imaginary Numbers
3:12
Imaginary i
3:51
Examples: Imaginary Number
4:22
Complex Numbers
7:23
Real Part and Imaginary Part
7:33
Examples: Complex Numbers
7:57
Equality
9:37
Example: Equal Complex Numbers
9:52
Addition and Subtraction
10:12
Examples: Adding Complex Numbers
10:25
Complex Plane
13:32
Horizontal Axis (Real)
13:49
Vertical Axis (Imaginary)
13:59
Example: Labeling
14:11
Multiplication
15:57
Example: FOIL Method
16:03
Division
18:37
Complex Conjugates
18:45
Conjugate Pairs
19:10
Example: Dividing Complex Numbers
20:00
Example 1: Simplify Complex Number
24:50
Example 2: Simplify Complex Number
27:56
Example 3: Multiply Complex Numbers
29:27
Example 3: Dividing Complex Numbers
31:48
Completing the Square

27m 11s

Intro
0:00
Square Root Property
0:12
Example: Perfect Square
0:38
Example: Perfect Square Trinomial
3:00
Completing the Square
4:39
Constant Term
4:50
Example: Complete the Square
5:04
Solve Equations
6:42
Add to Both Sides
6:59
Example: Complete the Square
7:07
Equations Where a Not Equal to 1
10:58
Divide by Coefficient
11:08
Example: Complete the Square
11:24
Complex Solutions
14:05
Real and Imaginary
14:14
Example: Complex Solution
14:35
Example 1: Square Root Property
18:31
Example 2: Complete the Square
19:15
Example 3: Complete the Square
20:40
Example 4: Complete the Square
23:56
Quadratic Formula and the Discriminant

22m 48s

Intro
0:00
Quadratic Formula
0:21
Standard Form
0:29
Example: Quadratic Formula
0:57
One Rational Root
3:00
Example: One Root
3:31
Complex Solutions
6:16
Complex Conjugate
6:28
Example: Complex Solution
7:15
Discriminant
9:42
Positive Discriminant
10:03
Perfect Square (Rational)
10:51
Not Perfect Square (2 Irrational)
11:27
Negative Discriminant
12:28
Zero Discriminant
12:57
Example 1: Quadratic Formula
13:50
Example 2: Quadratic Formula
16:03
Example 3: Quadratic Formula
19:00
Example 4: Discriminant
21:33
Analyzing the Graphs of Quadratic Functions

30m 7s

Intro
0:00
Vertex Form
0:12
H and K
0:32
Axis of Symmetry
0:36
Vertex
0:42
Example: Origin
1:00
Example: k = 2
2:12
Example: h = 1
4:27
Significance of Coefficient a
7:13
Example: |a| > 1
7:25
Example: |a| < 1
8:18
Example: |a| > 0
8:51
Example: |a| < 0
9:05
Writing Quadratic Equations in Vertex Form
10:22
Standard Form to Vertex Form
10:35
Example: Standard Form
11:02
Example: a Term Not 1
14:42
Example 1: Vertex Form
19:47
Example 2: Vertex Form
22:09
Example 3: Vertex Form
24:32
Example 4: Vertex Form
28:23
Graphing and Solving Quadratic Inequalities

27m 5s

Intro
0:00
Graphing Quadratic Inequalities
0:11
Test Point
0:18
Example: Quadratic Inequality
0:29
Solving Quadratic Inequalities
3:57
Example: Parameter
4:24
Example 1: Graph Inequality
11:16
Example 2: Solve Inequality
14:27
Example 3: Graph Inequality
19:14
Example 4: Solve Inequality
23:48
Section 6: Polynomial Functions
Properties of Exponents

19m 29s

Intro
0:00
Simplifying Exponential Expressions
0:09
Monomial Simplest Form
0:19
Negative Exponents
1:07
Examples: Simple
1:34
Properties of Exponents
3:06
Negative Exponents
3:13
Mutliplying Same Base
3:24
Dividing Same Base
3:45
Raising Power to a Power
4:33
Parentheses (Multiplying)
5:11
Parentheses (Dividing)
5:47
Raising to 0th Power
6:15
Example 1: Simplify Exponents
7:59
Example 2: Simplify Exponents
10:41
Example 3: Simplify Exponents
14:11
Example 4: Simplify Exponents
18:04
Operations on Polynomials

13m 27s

Intro
0:00
Adding and Subtracting Polynomials
0:13
Like Terms and Like Monomials
0:23
Examples: Adding Monomials
1:14
Multiplying Polynomials
3:40
Distributive Property
3:44
Example: Monomial by Polynomial
4:06
Example 1: Simplify Polynomials
5:47
Example 2: Simplify Polynomials
6:28
Example 3: Simplify Polynomials
8:38
Example 4: Simplify Polynomials
10:47
Dividing Polynomials

31m 11s

Intro
0:00
Dividing by a Monomial
0:13
Example: Numbers
0:26
Example: Polynomial by a Monomial
1:18
Long Division
2:28
Remainder Term
2:41
Example: Dividing with Numbers
3:04
Example: With Polynomials
5:01
Example: Missing Terms
7:58
Synthetic Division
11:44
Restriction
12:04
Example: Divisor in Form
12:20
Divisor in Synthetic Division
15:54
Example: Coefficient to 1
16:07
Example 1: Divide Polynomials
17:10
Example 2: Divide Polynomials
19:08
Example 3: Synthetic Division
21:42
Example 4: Synthetic Division
25:09
Polynomial Functions

22m 30s

Intro
0:00
Polynomial in One Variable
0:13
Leading Coefficient
0:27
Example: Polynomial
1:18
Degree
1:31
Polynomial Functions
2:57
Example: Function
3:13
Function Values
3:33
Example: Numerical Values
3:53
Example: Algebraic Expressions
5:11
Zeros of Polynomial Functions
5:50
Odd Degree
6:04
Even Degree
7:29
End Behavior
8:28
Even Degrees
9:09
Example: Leading Coefficient +/-
9:23
Odd Degrees
12:51
Example: Leading Coefficient +/-
13:00
Example 1: Degree and Leading Coefficient
15:03
Example 2: Polynomial Function
15:56
Example 3: Polynomial Function
17:34
Example 4: End Behavior
19:53
Analyzing Graphs of Polynomial Functions

33m 29s

Intro
0:00
Graphing Polynomial Functions
0:11
Example: Table and End Behavior
0:39
Location Principle
4:43
Zero Between Two Points
5:03
Example: Location Principle
5:21
Maximum and Minimum Points
8:40
Relative Maximum and Relative Minimum
9:16
Example: Number of Relative Max/Min
11:11
Example 1: Graph Polynomial Function
11:57
Example 2: Graph Polynomial Function
16:19
Example 3: Graph Polynomial Function
23:27
Example 4: Graph Polynomial Function
28:35
Solving Polynomial Functions

21m 10s

Intro
0:00
Factoring Polynomials
0:06
Greatest Common Factor (GCF)
0:25
Difference of Two Squares
1:14
Perfect Square Trinomials
2:07
General Trinomials
2:57
Grouping
4:32
Sum and Difference of Two Cubes
6:03
Examples: Two Cubes
6:14
Quadratic Form
8:22
Example: Quadratic Form
8:44
Example 1: Factor Polynomial
12:03
Example 2: Factor Polynomial
13:54
Example 3: Quadratic Form
15:33
Example 4: Solve Polynomial Function
17:24
Remainder and Factor Theorems

31m 21s

Intro
0:00
Remainder Theorem
0:07
Checking Work
0:22
Dividend and Divisor in Theorem
1:12
Example: f(a)
2:05
Synthetic Substitution
5:43
Example: Polynomial Function
6:15
Factor Theorem
9:54
Example: Numbers
10:16
Example: Confirm Factor
11:27
Factoring Polynomials
14:48
Example: 3rd Degree Polynomial
15:07
Example 1: Remainder Theorem
19:17
Example 2: Other Factors
21:57
Example 3: Remainder Theorem
25:52
Example 4: Other Factors
28:21
Roots and Zeros

31m 27s

Intro
0:00
Number of Roots
0:08
Not Nature of Roots
0:18
Example: Real and Complex Roots
0:25
Descartes' Rule of Signs
2:05
Positive Real Roots
2:21
Example: Positve
2:39
Negative Real Roots
5:44
Example: Negative
6:06
Finding the Roots
9:59
Example: Combination of Real and Complex
10:07
Conjugate Roots
13:18
Example: Conjugate Roots
13:50
Example 1: Solve Polynomial
16:03
Example 2: Solve Polynomial
18:36
Example 3: Possible Combinations
23:13
Example 4: Possible Combinations
27:11
Rational Zero Theorem

31m 16s

Intro
0:00
Equation
0:08
List of Possibilities
0:16
Equation with Constant and Leading Coefficient
1:04
Example: Rational Zero
2:46
Leading Coefficient Equal to One
7:19
Equation with Leading Coefficient of One
7:34
Example: Coefficient Equal to 1
8:45
Finding Rational Zeros
12:58
Division with Remainder Zero
13:32
Example 1: Possible Rational Zeros
14:20
Example 2: Possible Rational Zeros
16:02
Example 3: Possible Rational Zeros
19:58
Example 4: Find All Zeros
22:06
Section 7: Radical Expressions and Inequalities
Operations on Functions

34m 30s

Intro
0:00
Arithmetic Operations
0:07
Domain
0:16
Intersection
0:24
Denominator is Zero
0:49
Example: Operations
1:02
Composition of Functions
7:18
Notation
7:48
Right to Left
8:18
Example: Composition
8:48
Composition is Not Commutative
17:23
Example: Not Commutative
17:51
Example 1: Function Operations
20:55
Example 2: Function Operations
24:34
Example 3: Compositions
27:51
Example 4: Function Operations
31:09
Inverse Functions and Relations

22m 42s

Intro
0:00
Inverse of a Relation
0:14
Example: Ordered Pairs
0:56
Inverse of a Function
3:24
Domain and Range Switched
3:52
Example: Inverse
4:28
Procedure to Construct an Inverse Function
6:42
f(x) to y
6:42
Interchange x and y
6:59
Solve for y
7:06
Write Inverse f(x) for y
7:14
Example: Inverse Function
7:25
Example: Inverse Function 2
8:48
Inverses and Compositions
10:44
Example: Inverse Composition
11:46
Example 1: Inverse Relation
14:49
Example 2: Inverse of Function
15:40
Example 3: Inverse of Function
17:06
Example 4: Inverse Functions
18:55
Square Root Functions and Inequalities

30m 4s

Intro
0:00
Square Root Functions
0:07
Examples: Square Root Function
0:16
Example: Not Square Root Function
0:46
Radicand
1:12
Example: Restriction
1:31
Graphing Square Root Functions
3:42
Example: Graphing
3:49
Square Root Inequalities
8:47
Same Technique
9:00
Example: Square Root Inequality
9:20
Example 1: Graph Square Root Function
15:19
Example 2: Graph Square Root Function
18:03
Example 3: Graph Square Root Function
22:41
Example 4: Square Root Inequalities
25:37
nth Roots

20m 46s

Intro
0:00
Definition of the nth Root
0:07
Example: 5th Root
0:20
Example: 6th Root
0:51
Principal nth Root
1:39
Example: Principal Roots
2:06
Using Absolute Values
5:58
Example: Square Root
6:18
Example: 6th Root
8:40
Example: Negative
10:15
Example 1: Simplify Radicals
12:23
Example 2: Simplify Radicals
13:29
Example 3: Simplify Radicals
16:07
Example 4: Simplify Radicals
18:18
Operations with Radical Expressions

41m 11s

Intro
0:00
Properties of Radicals
0:16
Quotient Property
0:29
Example: Quotient
1:00
Example: Product Property
1:47
Simplifying Radical Expressions
3:24
Radicand No nth Powers
3:47
Radicand No Fractions
6:33
No Radicals in Denominator
7:16
Rationalizing Denominators
8:27
Example: Radicand nth Power
9:05
Conjugate Radical Expressions
11:47
Conjugates
12:07
Example: Conjugate Radical Expression
13:11
Adding and Subtracting Radicals
16:12
Same Index, Same Radicand
16:20
Example: Like Radicals
16:28
Multiplying Radicals
19:04
Distributive Property
19:10
Example: Multiplying Radicals
19:20
Example 1: Simplify Radical
24:11
Example 2: Simplify Radicals
28:43
Example 3: Simplify Radicals
32:00
Example 4: Simplify Radical
36:34
Rational Exponents

30m 45s

Intro
0:00
Definition 1
0:20
Example: Using Numbers
0:39
Example: Non-Negative
2:46
Example: Odd
3:34
Definition 2
4:32
Restriction
4:52
Example: Relate to Definition 1
5:04
Example: m Not 1
5:31
Simplifying Expressions
7:53
Multiplication
8:31
Division
9:29
Multiply Exponents
10:08
Raised Power
11:05
Zero Power
11:29
Negative Power
11:49
Simplified Form
13:52
Complex Fraction
14:16
Negative Exponents
14:40
Example: More Complicated
15:14
Example 1: Write as Radical
19:03
Example 2: Write with Rational Exponents
20:40
Example 3: Complex Fraction
22:09
Example 4: Complex Fraction
26:22
Solving Radical Equations and Inequalities

31m 27s

Intro
0:00
Radical Equations
0:11
Variables in Radicands
0:22
Example: Radical Equation
1:06
Example: Complex Equation
2:42
Extraneous Roots
7:21
Squaring Technique
7:35
Double Check
7:44
Example: Extraneous
8:21
Eliminating nth Roots
10:04
Isolate and Raise Power
10:14
Example: nth Root
10:27
Radical Inequalities
11:27
Restriction: Index is Even
11:53
Example: Radical Inequality
12:29
Example 1: Solve Radical Equation
15:41
Example 2: Solve Radical Equation
17:44
Example 3: Solve Radical Inequality
20:24
Example 4: Solve Radical Equation
24:34
Section 8: Rational Equations and Inequalities
Multiplying and Dividing Rational Expressions

40m 54s

Intro
0:00
Simplifying Rational Expressions
0:22
Algebraic Fraction
0:29
Examples: Rational Expressions
0:49
Example: GCF
1:33
Example: Simplify Rational Expression
2:26
Factoring -1
4:04
Example: Simplify with -1
4:19
Multiplying and Dividing Rational Expressions
6:59
Multiplying and Dividing
7:28
Example: Multiplying Rational Expressions
8:36
Example: Dividing Rational Expressions
11:20
Factoring
14:01
Factoring Polynomials
14:19
Example: Factoring
14:35
Complex Fractions
18:22
Example: Numbers
18:37
Example: Algebraic Complex Fractions
19:25
Example 1: Simplify Rational Expression
25:56
Example 2: Simplify Rational Expression
29:34
Example 3: Simplify Rational Expression
31:39
Example 4: Simplify Rational Expression
37:50
Adding and Subtracting Rational Expressions

55m 4s

Intro
0:00
Least Common Multiple (LCM)
0:27
Examples: LCM of Numbers
0:43
Example: LCM of Polynomials
4:02
Adding and Subtracting
7:55
Least Common Denominator (LCD)
8:07
Example: Numbers
8:17
Example: Rational Expressions
11:03
Equivalent Fractions
15:22
Simplifying Complex Fractions
21:19
Example: Previous Lessons
21:36
Example: More Complex
22:53
Example 1: Find LCM
28:30
Example 2: Add Rational Expressions
31:44
Example 3: Subtract Rational Expressions
39:18
Example 4: Simplify Rational Expression
38:26
Graphing Rational Functions

57m 13s

Intro
0:00
Rational Functions
0:18
Restriction
0:34
Example: Rational Function
0:51
Breaks in Continuity
2:52
Example: Continuous Function
3:10
Discontinuities
3:30
Example: Excluded Values
4:37
Graphs and Discontinuities
5:02
Common Binomial Factor (Hole)
5:08
Example: Common Factor
5:31
Asymptote
10:06
Example: Vertical Asymptote
11:08
Horizontal Asymptotes
20:00
Example: Horizontal Asymptote
20:25
Example 1: Holes and Vertical Asymptotes
26:12
Example 2: Graph Rational Faction
28:35
Example 3: Graph Rational Faction
39:23
Example 4: Graph Rational Faction
47:28
Direct, Joint, and Inverse Variation

20m 21s

Intro
0:00
Direct Variation
0:07
Constant of Variation
0:25
Graph of Constant Variation
1:26
Slope is Constant k
1:35
Example: Straight Lines
1:41
Joint Variation
2:48
Three Variables
2:52
Inverse Variation
3:38
Rewritten Form
3:52
Examples in Biology
4:22
Graph of Inverse Variation
4:51
Asymptotes are Axes
5:12
Example: Inverse Variation
5:40
Proportions
10:11
Direct Variation
10:25
Inverse Variation
11:32
Example 1: Type of Variation
12:42
Example 2: Direct Variation
14:13
Example 3: Joint Variation
16:24
Example 4: Graph Rational Faction
18:50
Solving Rational Equations and Inequalities

55m 14s

Intro
0:00
Rational Equations
0:15
Example: Algebraic Fraction
0:26
Least Common Denominator
0:49
Example: Simple Rational Equation
1:22
Example: Solve Rational Equation
5:40
Extraneous Solutions
9:31
Doublecheck
10:00
No Solution
10:38
Example: Extraneous
10:44
Rational Inequalities
14:01
Excluded Values
14:31
Solve Related Equation
14:49
Find Intervals
14:58
Use Test Values
15:25
Example: Rational Inequality
15:51
Example: Rational Inequality 2
17:07
Example 1: Rational Equation
28:50
Example 2: Rational Equation
33:51
Example 3: Rational Equation
38:19
Example 4: Rational Inequality
46:49
Section 9: Exponential and Logarithmic Relations
Exponential Functions

35m 58s

Intro
0:00
What is an Exponential Function?
0:12
Restriction on b
0:31
Base
0:46
Example: Exponents as Bases
0:56
Variables as Exponents
1:12
Example: Exponential Function
1:50
Graphing Exponential Functions
2:33
Example: Using Table
2:49
Properties
11:52
Continuous and One to One
12:00
Domain is All Real Numbers
13:14
X-Axis Asymptote
13:55
Y-Intercept
14:02
Reflection Across Y-Axis
14:31
Growth and Decay
15:06
Exponential Growth
15:10
Real Life Examples
15:41
Example: Growth
15:52
Example: Decay
16:12
Real Life Examples
16:30
Equations
17:32
Bases are Same
18:05
Examples: Variables as Exponents
18:20
Inequalities
21:29
Property
21:51
Example: Inequality
22:37
Example 1: Graph Exponential Function
24:05
Example 2: Growth or Decay
27:50
Example 3: Exponential Equation
29:31
Example 4: Exponential Inequality
32:54
Logarithms and Logarithmic Functions

45m 54s

Intro
0:00
What are Logarithms?
0:08
Restrictions
0:15
Written Form
0:26
Logarithms are Exponents
0:52
Example: Logarithms
1:49
Logarithmic Functions
5:14
Same Restrictions
5:30
Inverses
5:53
Example: Logarithmic Function
6:24
Graph of the Logarithmic Function
9:20
Example: Using Table
9:35
Properties
15:09
Continuous and One to One
15:14
Domain
15:36
Range
15:56
Y-Axis is Asymptote
16:02
X Intercept
16:12
Inverse Property
16:57
Compositions of Functions
17:10
Equations
18:30
Example: Logarithmic Equation
19:13
Inequalities
20:36
Properties
20:47
Example: Logarithmic Inequality
21:40
Equations with Logarithms on Both Sides
24:43
Property
24:51
Example: Both Sides
25:23
Inequalities with Logarithms on Both Sides
26:52
Property
27:02
Example: Both Sides
28:05
Example 1: Solve Log Equation
31:52
Example 2: Solve Log Equation
33:53
Example 3: Solve Log Equation
36:15
Example 4: Solve Log Inequality
39:19
Properties of Logarithms

28m 43s

Intro
0:00
Product Property
0:08
Example: Product
0:46
Quotient Property
2:40
Example: Quotient
2:59
Power Property
3:51
Moved Exponent
4:07
Example: Power
4:37
Equations
5:15
Example: Use Properties
5:58
Example 1: Simplify Log
11:17
Example 2: Single Log
15:54
Example 3: Solve Log Equation
18:48
Example 4: Solve Log Equation
22:13
Common Logarithms

25m 23s

Intro
0:00
What are Common Logarithms?
0:10
Real World Applications
0:16
Base Not Written
0:27
Example: Base 10
0:39
Equations
1:47
Example: Same Base
1:56
Example: Different Base
2:37
Inequalities
6:07
Multiplying/Dividing Inequality
6:21
Example: Log Inequality
6:54
Change of Base
12:45
Base 10
13:24
Example: Change of Base
14:05
Example 1: Log Equation
15:21
Example 2: Common Logs
17:13
Example 3: Log Equation
18:22
Example 4: Log Inequality
21:52
Base e and Natural Logarithms

21m 14s

Intro
0:00
Number e
0:09
Natural Base
0:21
Growth/Decay
0:33
Example: Exponential Function
0:53
Natural Logarithms
1:11
ln x
1:19
Inverse and Identity Function
1:39
Example: Inverse Composition
1:55
Equations and Inequalities
4:39
Extraneous Solutions
5:30
Examples: Natural Log Equations
5:48
Example 1: Natural Log Equation
9:08
Example 2: Natural Log Equation
10:37
Example 3: Natural Log Inequality
16:54
Example 4: Natural Log Inequality
18:16
Exponential Growth and Decay

24m 30s

Intro
0:00
Decay
0:17
Decreases by Fixed Percentage
0:23
Rate of Decay
0:56
Example: Finance
1:34
Scientific Model of Decay
3:37
Exponential Decay
3:45
Radioactive Decay
4:13
Example: Half Life
5:33
Growth
9:06
Increases by Fixed Percentage
9:18
Example: Finance
10:09
Scientific Model of Growth
11:35
Population Growth
12:04
Example: Growth
12:20
Example 1: Computer Price
14:00
Example 2: Stock Price
15:46
Example 3: Medicine Disintegration
19:10
Example 4: Population Growth
22:33
Section 10: Conic Sections
Midpoint and Distance Formulas

32m 42s

Intro
0:00
Midpoint Formula
0:15
Example: Midpoint
0:30
Distance Formula
2:30
Example: Distance
2:52
Example 1: Midpoint and Distance
4:58
Example 2: Midpoint and Distance
8:07
Example 3: Median Length
18:51
Example 4: Perimeter and Area
23:36
Parabolas

41m 27s

Intro
0:00
What is a Parabola?
0:20
Definition of a Parabola
0:29
Focus
0:59
Directrix
1:15
Axis of Symmetry
3:08
Vertex
3:33
Minimum or Maximum
3:44
Standard Form
4:59
Horizontal Parabolas
5:08
Vertex Form
5:19
Upward or Downward
5:41
Example: Standard Form
6:06
Graphing Parabolas
8:31
Shifting
8:51
Example: Completing the Square
9:22
Symmetry and Translation
12:18
Example: Graph Parabola
12:40
Latus Rectum
17:13
Length
18:15
Example: Latus Rectum
18:35
Horizontal Parabolas
18:57
Not Functions
20:08
Example: Horizontal Parabola
21:21
Focus and Directrix
24:11
Horizontal
24:48
Example 1: Parabola Standard Form
25:12
Example 2: Graph Parabola
30:00
Example 3: Graph Parabola
33:13
Example 4: Parabola Equation
37:28
Circles

21m 3s

Intro
0:00
What are Circles?
0:08
Example: Equidistant
0:17
Radius
0:32
Equation of a Circle
0:44
Example: Standard Form
1:11
Graphing Circles
1:47
Example: Circle
1:56
Center Not at Origin
3:07
Example: Completing the Square
3:51
Example 1: Equation of Circle
6:44
Example 2: Center and Radius
11:51
Example 3: Radius
15:08
Example 4: Equation of Circle
16:57
Ellipses

46m 51s

Intro
0:00
What Are Ellipses?
0:11
Foci
0:23
Properties of Ellipses
1:43
Major Axis, Minor Axis
1:47
Center
1:54
Length of Major Axis and Minor Axis
3:21
Standard Form
5:33
Example: Standard Form of Ellipse
6:09
Vertical Major Axis
9:14
Example: Vertical Major Axis
9:46
Graphing Ellipses
12:51
Complete the Square and Symmetry
13:00
Example: Graphing Ellipse
13:16
Equation with Center at (h, k)
19:57
Horizontal and Vertical
20:14
Difference
20:27
Example: Center at (h, k)
20:55
Example 1: Equation of Ellipse
24:05
Example 2: Equation of Ellipse
27:57
Example 3: Equation of Ellipse
32:32
Example 4: Graph Ellipse
38:27
Hyperbolas

38m 15s

Intro
0:00
What are Hyperbolas?
0:12
Two Branches
0:18
Foci
0:38
Properties
2:00
Transverse Axis and Conjugate Axis
2:06
Vertices
2:46
Length of Transverse Axis
3:14
Distance Between Foci
3:31
Length of Conjugate Axis
3:38
Standard Form
5:45
Vertex Location
6:36
Known Points
6:52
Vertical Transverse Axis
7:26
Vertex Location
7:50
Asymptotes
8:36
Vertex Location
8:56
Rectangle
9:28
Diagonals
10:29
Graphing Hyperbolas
12:58
Example: Hyperbola
13:16
Equation with Center at (h, k)
16:32
Example: Center at (h, k)
17:21
Example 1: Equation of Hyperbola
19:20
Example 2: Equation of Hyperbola
22:48
Example 3: Graph Hyperbola
26:05
Example 4: Equation of Hyperbola
36:29
Conic Sections

18m 43s

Intro
0:00
Conic Sections
0:16
Double Cone Sections
0:24
Standard Form
1:27
General Form
1:37
Identify Conic Sections
2:16
B = 0
2:50
X and Y
3:22
Identify Conic Sections, Cont.
4:46
Parabola
5:17
Circle
5:51
Ellipse
6:31
Hyperbola
7:10
Example 1: Identify Conic Section
8:01
Example 2: Identify Conic Section
11:03
Example 3: Identify Conic Section
11:38
Example 4: Identify Conic Section
14:50
Solving Quadratic Systems

47m 4s

Intro
0:00
Linear Quadratic Systems
0:22
Example: Linear Quadratic System
0:45
Solutions
2:49
Graphs of Possible Solutions
3:10
Quadratic Quadratic System
4:10
Example: Elimination
4:21
Solutions
11:39
Example: 0, 1, 2, 3, 4 Solutions
11:50
Systems of Quadratic Inequalities
12:48
Example: Quadratic Inequality
13:09
Example 1: Solve Quadratic System
21:42
Example 2: Solve Quadratic System
29:13
Example 3: Solve Quadratic System
35:02
Example 4: Solve Quadratic Inequality
40:29
Section 11: Sequences and Series
Arithmetic Sequences

21m 16s

Intro
0:00
Sequences
0:10
General Form of Sequence
0:16
Example: Finite/Infinite Sequences
0:33
Arithmetic Sequences
0:28
Common Difference
2:41
Example: Arithmetic Sequence
2:50
Formula for the nth Term
3:51
Example: nth Term
4:32
Equation for the nth Term
6:37
Example: Using Formula
6:56
Arithmetic Means
9:47
Example: Arithmetic Means
10:16
Example 1: nth Term
12:38
Example 2: Arithmetic Means
13:49
Example 3: Arithmetic Means
16:12
Example 4: nth Term
18:26
Arithmetic Series

21m 36s

Intro
0:00
What are Arithmetic Series?
0:11
Common Difference
0:28
Example: Arithmetic Sequence
0:43
Example: Arithmetic Series
1:09
Finite/Infinite Series
1:36
Sum of Arithmetic Series
2:27
Example: Sum
3:21
Sigma Notation
5:53
Index
6:14
Example: Sigma Notation
7:14
Example 1: First Term
9:00
Example 2: Three Terms
10:52
Example 3: Sum of Series
14:14
Example 4: Sum of Series
18:13
Geometric Sequences

23m 3s

Intro
0:00
Geometric Sequences
0:11
Common Difference
0:38
Common Ratio
1:08
Example: Geometric Sequence
2:38
nth Term of a Geometric Sequence
4:41
Example: nth Term
4:56
Geometric Means
6:51
Example: Geometric Mean
7:09
Example 1: 9th Term
12:04
Example 2: Geometric Means
15:18
Example 3: nth Term
18:32
Example 4: Three Terms
20:59
Geometric Series

22m 43s

Intro
0:00
What are Geometric Series?
0:11
List of Numbers
0:24
Example: Geometric Series
1:12
Sum of Geometric Series
2:16
Example: Sum of Geometric Series
2:41
Sigma Notation
4:21
Lower Index, Upper Index
4:38
Example: Sigma Notation
4:57
Another Sum Formula
6:08
Example: n Unknown
6:28
Specific Terms
7:41
Sum Formula
7:56
Example: Specific Term
8:11
Example 1: Sum of Geometric Series
10:02
Example 2: Sum of 8 Terms
14:15
Example 3: Sum of Geometric Series
18:23
Example 4: First Term
20:16
Infinite Geometric Series

18m 32s

Intro
0:00
What are Infinite Geometric Series
0:10
Example: Finite
0:29
Example: Infinite
0:51
Partial Sums
1:09
Formula
1:37
Sum of an Infinite Geometric Series
2:39
Convergent Series
2:58
Example: Sum of Convergent Series
3:28
Sigma Notation
7:31
Example: Sigma
8:17
Repeating Decimals
8:42
Example: Repeating Decimal
8:53
Example 1: Sum of Infinite Geometric Series
12:15
Example 2: Repeating Decimal
13:24
Example 3: Sum of Infinite Geometric Series
15:14
Example 4: Repeating Decimal
16:48
Recursion and Special Sequences

14m 34s

Intro
0:00
Fibonacci Sequence
0:05
Background of Fibonacci
0:23
Recursive Formula
0:37
Fibonacci Sequence
0:52
Example: Recursive Formula
2:18
Iteration
3:49
Example: Iteration
4:30
Example 1: Five Terms
7:08
Example 2: Three Terms
9:00
Example 3: Five Terms
10:38
Example 4: Three Iterates
12:41
Binomial Theorem

48m 30s

Intro
0:00
Pascal's Triangle
0:06
Expand Binomial
0:13
Pascal's Triangle
4:26
Properties
6:52
Example: Properties of Binomials
6:58
Factorials
9:11
Product
9:28
Example: Factorial
9:45
Binomial Theorem
11:08
Example: Binomial Theorem
13:48
Finding a Specific Term
18:36
Example: Specific Term
19:26
Example 1: Expand
24:39
Example 2: Fourth Term
30:26
Example 3: Five Terms
36:13
Example 4: Three Iterates
45:07
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Arithmetic Series

  • There are two formulas for the sum. Use one if you are given d and the other if you are given an. In both cases, you will be given the values of a1 and n.
  • In some problems, you must first use the formula for the sum and then the formula for the nth term.

Arithmetic Series

Find the first term for the arithmetic series with
d = 4, n = 6, and S6 = 156
  • You can use the formula Sn = [n/2]( 2a1 + (n − 1)d )
  • Plug in the values for d, n and Sn
  • Sn = [n/2]( 2a1 + (n − 1)d )
  • 156 = [6/2]( 2a1 + (6 − 1)4 )
  • 156 = 3( 2a1 + (5)4 )
  • 156 = 3( 2a1 + 20 )
  • 156 = 6a1 + 60
  • 96 = 6a1
a1 = 16
Find the first term for the arithmetic series with
d = − 6, n = 9, and S9 = − 288
  • You can use the formula Sn = [n/2]( 2a1 + (n − 1)d )
  • Plug in the values for d, n and Sn
  • Sn = [n/2]( 2a1 + (n − 1)d )
  • − 288 = [9/2]( 2a1 + (9 − 1)( − 6) )
  • − 576 = 9( 2a1 + (8)( − 6) )
  • − 576 = 9( 2a1 − 48 )
  • − 576 = 18a1 − 432
  • − 144 = 18a1
a1 = − 8
Find the first term for the arithmetic series with
d = 5, n = 10, and S9 = 325
  • You can use the formula Sn = [n/2]( 2a1 + (n − 1)d )
  • Plug in the values for d, n and Sn
  • Sn = [n/2]( 2a1 + (n − 1)d )
  • 325 = [10/2]( 2a1 + (10 − 1)(5) )
  • 325 = 5( 2a1 + (9)(5) )
  • 325 = 5( 2a1 + 45 )
  • 325 = 10a1 + 225
  • 100 = 10a1
a1 = 10
Find the first three terms for the arithmetic series with
n = 8, an = 46, and Sn = 172
  • You can use the formula Sn = [n/2]( a1 + an );Sn = [n/2]( 2a1 + (n − 1)d )
  • Step 1: Plug in the values for n, an and Sn to find the first term.
  • Sn = [n/2]( a1 + an )
  • 172 = [8/2]( a1 + 46 )
  • 172 = 4(a1 + 46)
  • 172 = 4a1 + 184
  • − 12 = 4a1
  • a1 = − 3
  • Step 2 - Use the equation Sn = [n/2]( 2a1 + (n − 1)d ) to find the common difference
  • 172 = [8/2]( 2( − 3) + (8 − 1)d )
  • 172 = 4( − 6 + 7d )
  • 172 = − 24 + 28d
  • 196 = 28d
  • d = 7
  • Step 3 - Find the next three terms using the 1st term and common difference
  • a1 = − 3
  • a2 =
  • a3 =
  • a4 =
a2 = 4a3 = 11a4 = 18
Find the first three terms for the arithmetic series with
n = 40, an = 175, and Sn = 3880
  • You can use the formula Sn = [n/2]( a1 + an );
  • Step 1: Plug in the values for n, an and Sn to find the first term.
  • Sn = [n/2]( a1 + an )
  • 3880 = [40/2]( a1 + 175 )
  • 3880 = 20(a1 + 175)
  • 3880 = 20a1 + 3500
  • 380 = 20a1
  • a1 = 19
  • Step 2 - Use the equation Sn = [n/2]( 2a1 + (n − 1)d ) to find the common difference
  • 3880 = [40/2]( 2(19) + (40 − 1)d )
  • 3880 = 20( 38 + 39d )
  • 3880 = 760 + 780d
  • 3120 = 780d
  • d = 4
  • Step 3 - Find the next three terms using the 1st term and common difference
  • a1 = 19
  • a2 =
  • a3 =
  • a4 =
a2 = 23a3 = 27a4 = 31
Find the first three terms for the arithmetic series with
n = 30, an = 74, and Sn = 1350
  • You can use the formula Sn = [n/2]( a1 + an );Sn = [n/2]( 2a1 + (n − 1)d )
  • Step 1: Plug in the values for n, an and Sn to find the first term.
  • Sn = [n/2]( a1 + an )
  • 1350 = [30/2]( a1 + 74 )
  • 1350 = 15(a1 + 74)
  • 1350 = 15a1 + 1110
  • 240 = 15a1
  • a1 = 16
  • Step 2 - Use the equation Sn = [n/2]( 2a1 + (n − 1)d ) to find the common difference
  • 1350 = [30/2]( 2(16) + (30 − 1)d )
  • 1350 = 15( 32 + 29d )
  • 1350 = 480 + 435d
  • 870 = 435d
  • d = 2
  • Step 3 - Find the next three terms using the 1st term and common difference
  • a1 = 16
  • a2 =
  • a3 =
  • a4 =
a2 = 18a3 = 20a4 = 22
Find the sum of the series
∑x = 16 (3x − 11 )
  • ∑x = 16 (3x − 11 ) = (3(1) − 11) + (3(2) − 11) + (3(3) − 11) + (3(4) − 11) + (3(5) − 11) + (3(6) − 11) =
∑x = 16 (3x − 11 ) = − 3
Find the sum of the series
∑x = 110 (6x − 4 )
  • ∑x = 110 (6x − 4 ) = (6( ) − 4) + (6( ) − 4) + (6( ) − 4) + (6( ) − 4) + (6( ) − 4) + (6( ) − 4) + (6( ) − 4) + (6( ) − 4) + (6( ) − 4) + (6( ) − 4)
  • ∑x = 110 (6x − 4 ) = (6(1) − 4) + (6(2) − 4) + (6(3) − 4) + (6(4) − 4) + (6(5) − 4) + (6(6) − 4) + (6(7) − 4) + (6(8) − 4) + (6(9) − 4) + (6(10) − 4)
∑x = 110 (6x − 4 ) = 290
Find the sum of the series
∑x = 19 (10x − 13 )
  • ∑x = 19 (10x − 13 ) = (10() − 13) + (10() − 13) + (10() − 13) + (10() − 13) + (10() − 13) + (10() − 13) + (10() − 13) + (10() − 13) + (10() − 13)
  • ∑x = 19 (10x − 13 ) = (10(1) − 13) + (10(2) − 13) + (10(3) − 13) + (10(4) − 13) + (10(5) − 13) + (10(6) − 13) + (10(7) − 13) + (10(8) − 13) + (10(9) − 13)
∑x = 19 (10x − 13 ) = 333
Find the sum of the series
∑x = 16 (10x − 11 )
  • ∑x = 16 (10x − 11 ) = + (10() − 11) + (10() − 11) + (10() − 11) + (10() − 11) + (10() − 11) + (10() − 11)
  • ∑x = 16 (10x − 11 ) = (10(1) − 11) + (10(2) − 11) + (10(3) − 11) + (10(4) − 11) + (10(5) − 11) + (10(6) − 11)
∑x = 16 (10x − 11 ) = 144

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

Arithmetic Series

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • What are Arithmetic Series? 0:11
    • Common Difference
    • Example: Arithmetic Sequence
    • Example: Arithmetic Series
    • Finite/Infinite Series
  • Sum of Arithmetic Series 2:27
    • Example: Sum
  • Sigma Notation 5:53
    • Index
    • Example: Sigma Notation
  • Example 1: First Term 9:00
  • Example 2: Three Terms 10:52
  • Example 3: Sum of Series 14:14
  • Example 4: Sum of Series 18:13

Transcription: Arithmetic Series

Welcome to Educator.com.0000

In a previous lesson, we talked about arithmetic sequences; and in this lesson, we will continue on with the discussion, to discuss arithmetic s0ries.0002

First of all, what are arithmetic series? An arithmetic series is the sum of the terms of an arithmetic sequence.0012

So, I will briefly review arithmetic sequences; but if you need a complete review of that,0019

go and check out the previous lesson, and then move on to arithmetic series.0024

Recall that a series is a list of numbers in a particular order; and each term in the series is related to the previous one by a constant called the common difference.0028

A typical arithmetic sequence would be something like 20, 40, 60, 80, 100.0043

Taking 40 - 20 or 60 - 40, I get a common difference of 20, which means that, to go from one term to the next, I add 20.0051

So, this is the sequence: it is a list of numbers; the series is actually a sum of numbers,0063

so I am going to add a + between each number: 20 + 40 + 60 + 80 + 100.0071

So, an arithmetic series is in the general form: first term, a1 + a2 + a3, and on and on until the last term, an.0082

This is a finite arithmetic series, because there is a specific endpoint; there is a finite number of terms.0097

As with arithmetic sequences (those could be finite or infinite), you could have finite or infinite arithmetic series.0106

So, looking at a different sequence: 100, 200, 300, 400, and then the ellipses to tell me that this is an infinite sequence:0115

I could have a corresponding series, 100 + 200 + 300 + 400 + ...on and on; this is an infinite arithmetic series.0127

There is a formula (or actually, two formulas) here to allow you to find the sum of an arithmetic series.0148

Sometimes you will be asked to find the sum of the first n terms of an arithmetic series--maybe the first 17 terms or the first 10 terms.0154

And that would be s17 or s10.0162

There are two formulas: the first one involves taking the number of terms, dividing it by 2, and then multiplying it0166

by 2 times the first term, plus the quantity (n - 1) times the common difference.0173

The second term is the sum of the first n terms: again, it equals the number of terms divided by 2.0179

But this time, you are going to multiply that by the sum of the first and last terms.0185

So, you can see the difference: you use the first formula if you know the common difference, and the second formula if you know the last term.0190

So, you have to choose a formula, depending on what you know.0199

For example, let's say that a series has, as its first term, a 5: 5 is the first term.0202

And it has a common difference of 4, and you are asked to find the sum of the first 10 terms, s10.0211

I am looking, and I see that I have the common difference; so I am going to go ahead and use the first formula.0220

And let's go ahead and do that: s10 = 10, because, since I am finding the sum of the first 10 terms,0229

n = 10, divided by 2, and that is times 2 times 5 (the first term), plus n (which is 10) minus 1, times the common difference (which is 4).0241

So, s10 = 5(10) +...10 - 1 is, of course, 9, times 4.0253

Therefore, s10 = 5 times...9 times 4 is 36; 36 + 10 is 46.0262

And if you multiply that out, or use your calculator, you will find that the sum of the first 10 terms0272

in a series with this first term and this common difference is actually 230.0276

So, that is one example; in another example, perhaps you are asked the sum of the 16 terms in a series,0283

if the first term is 20, and the last term of the ones we are trying to find, a16, is equal to -10.0292

Well, we are going to use the second formula, because we have the first and last terms, but we don't have the common difference.0305

So, s16 = n; in this case, I am looking for the sum of 16 terms, so n is 160311

(I am going to put that right here--n is 16); that is going to give me 16/2, times the first term, which is 20, plus that last term, which is -10.0323

So, this is s16 = 8, times 20 - 10, which is 10; that gives me 80 as the sum of the first 16 terms.0336

You need to learn both formulas and simply know when to use a particular one.0348

Sigma notation: a series can be written in a concise form, using what is called sigma notation.0353

And what sigma refers to is the Greek letter Σ; and in this case, sigma means "sum."0359

So, what this symbol means, in the context in which we are using it, is "sum."0366

And you will see it written something like this: you will see a letter here, called the index (that variable is called the index).0371

And we have been working with n, but often i is used; it could be n; it could be i; it could be k; it could be something else.0378

i = 1; I am just giving an example; and let's say, up here, the upper index is going to be 10.0385

And then, there will be the formula for the general term, which we talked about earlier on,0393

when we talked about arithmetic sequences and the formula for an that is particular to a sequence.0399

So, if you need to go ahead and review that, it is in the previous lecture.0406

This is the formula for the general term of the sequence.0409

And this is read as "the sum of an as i goes from 1 to 10."0412

So, it is the sum of the terms that are found, using this particular formula, when you plug in 1, then 2, then 3, and up through 10.0420

That is in general; let's talk about a specific example with a specific formula.0429

You could have another arithmetic series, written in sigma notation, where n goes from 5 to 12.0435

So, n goes from 5 to 12; so I am going to start with 5, and I am going to end with 12.0448

That means that there are actually 8 terms; there are 8 terms in the series, because it is 5 through 12, inclusive.0454

And the formula, we are going to say, is n + 3: so the formula to find a particular term, an, is n + 3.0460

I could find the terms, then, by saying that for the first term, a1, I am going to use 5 as my n, so it equals 5 + 3; therefore, it is 8.0472

For a2, I am then going to go to 6: so that is going to be 6 + 3; that is 9.0485

a3 =...then I am going to go to the next value, which is 7: 7 + 3 = 10; and on up.0497

So, if I wanted to find the last term, a8 (because there are 8 terms), then I would put in 12; and 12 + 3 is 15.0512

There would be terms in between here, of course.0524

This is just a concise way of writing a series; and we have already talked about how to work with these series, and what the different terms mean, and formulas.0527

But now, this is just a different way of writing them.0539

We need to find the first term of the arithmetic series with a common difference of 3.5 and equal to 20,0541

and the sum of the terms, s20, equaling 1005.0547

Since we know the common difference, we can use this formula: sn = n/2, times 2 times the first term, plus (n - 1)d.0553

Except, in this case, I am not looking for the sum: I have the sum; I am looking for the first term.0566

Therefore, 1005 = 20/2, times 2 times a1, plus n (n is given as 20), minus 1, times the common difference of 3.5.0571

This gives me 1005 = 10(2a1) + 19(3.5).0589

19 times 3.5 is actually 66.5; therefore, 1005 equals 10 times 2a1, which is 20a1, plus 10(66.5), which is 665.0601

1005; subtract 665 from both sides to get 340 = 20 times that first term.0622

Divide both terms by 20, and you will get that the first term is equal to 17.0631

So, we use this formula for the sum of the series; but in this case, we were looking for the first term.0636

We had the sum; we were looking for the first term.0646

And the solution is that the first term is 17.0647

In the second problem, we are asked to find the first three terms of the arithmetic series with n = 17, an = 103, sn = 1102.0653

In order to find the first three terms, I need to find the common difference.0666

And I also need to find the first term; I need the first term, and then I need the common difference, to find the second and third terms.0670

I can use the formula sn = n/2 times the first term plus the last term, because I don't have the common difference--I am looking for it.0680

But what I do have is an, so my first step is going to be to find the first term.0691

The sum is 1102; n is 19; I don't know the first term; and I know that an is 103.0698

I am going to multiply both sides by 2 to get 2204 = 19 times this, a1 + 103.0710

I am then going to divide both sides by 19, and that comes out to 116 = the first term, plus 103.0721

And then, I just subtract 103 from both sides; and now I have my first term.0729

So, I am asked to find the first three terms: the first term is 13.0734

To find the next two terms, I find the common difference.0737

What I am going to do is switch to the other formula: that other formula, sn,0740

equals n/2 times 2a1, the first term, plus n - 1 times the common difference.0744

I found the first term; now that I have that, I can find the common difference, because I can put the first term in here.0754

So again, sn is 1102; n is 19; and this gives me 2 times 13, plus I have an n of 19 - 1, and I am looking for the common difference.0761

I am going to multiply, again, 1102 times 2 to get 2204 = 19...2 times 13 is 26, plus 19 - 1...that gives me 18d.0775

I am going to divide both sides by 19 to get 116 = 26 + 18d.0790

Subtracting 26 from both sides gives me 90 = 18d; the final step is to divide both sides by 18 to get a common difference of 5.0798

I have a1 is 13; I am going to take 13 + the common difference of 5 to get the second term (that is 18).0810

So, a2 is 18; the third term--I am going to take 18, and I am going to add 5 to that to yield 23.0823

So, I was asked to find the first three terms, and I did that by first using this formula to find the first term,0836

then going to the other formula and finding the common difference to get 13, 18, and 23 as my solutions.0843

The third example: I am asked to find the sum of the series 6 + 11 + 16 + 21, and on and on, with the last term of 126.0854

That means that what I have is the first term and the last term.0867

I am going to find the sum using this formula, because I can figure out my common difference.0874

So, I am going to use the formula n/2, times 2a1, plus a - 1, times d.0884

I could easily find the common difference, because I know I will just take 11 - 6, so I have a common difference of 5.0896

Looking at this formula, the only issue is going to be that I don't know n.0904

But I can figure out n; and that is because I have another formula.0910

Recall the formula for the general term: we discussed this in the lecture on arithmetic sequences.0917

an equals the first term, plus n - 1, times the common difference.0926

an is 126; so that shouldn't be 16--that is 126: an is 126, and I have the first term equal to 6; and I am solving for n.0937

I know that the common difference is 5.0956

126...and then I am subtracting 6 from both sides: that gives me 120 = (n - 1)5, so 120 = 5n - 5.0959

Adding 5 to both sides gives me 125 = 5n; 125/5 is 25, so I have n = 25.0970

Now, again, I am asked to find the sum of the series; and I can use this formula, because I now have n; I have the first term; and I have the common difference.0982

So, let's go ahead and use that: it is actually s25 = n, which is 25, divided by 2, times 2 times that first term0990

(which is 6), plus (n - 1) (n is 25, minus 1), times the common difference of 5.1000

Now, it is just a matter of simplifying: this is going to give me 25/2 times 12, plus 24 times 5.1009

So, the sum equals 25/2, times 12; and then if you multiply out 24 times 5, you will get 120.1022

Therefore, the sum equals 25/2; 120 + 12 is going to give you 132.1033

Now that I have gotten it down to this point, I can simplify,1050

because this is going to give me 132/2, times 25; so that is s25 = 25 times 66.1054

You can multiply it out; or it is a good time to use your calculator to find that the sum equals 1650.1065

So again, in order to use this formula, I had my common difference.1074

I didn't have n; I solved for n; n equals 25.1077

I went back in, substituted those values in, and then came out with s25; the sum of this series is 1650.1081

Example 4: we are working with sigma notation, so you need to know how to read this notation.1093

And I am going to start with 4 and end with 14; and I can find the first term, because I am also given the formula for a general term in this series.1098

So, to find the sum of the series, let's just start out by finding the first term, because we know we are going to need that.1116

a1...we are going to begin with 4, so for the first term, n is going to equal 4.1123

It is 2 times 4, minus 3; a1 = 8 - 3, so the first term is going to be equal to 5.1131

Now, recall that we have two formulas that we can use to find the sum.1143

We have one formula that involves knowing the common difference.1146

We have another formula that requires us to know the first term and the last term.1150

I found the first term; since I know that, for the last term, n = 14, I can find that, as well.1156

And what that means is that I can use this formula: that the sum of the series is going to be equal to n/2, times the first term, plus the last term.1162

Therefore, let me find the last term: an = 2(14) - 3; this is going to give me an = 28 - 3, so the last term equals 25.1172

Now, what is n? Well, this is telling me that the number of terms...I would have to take each number from 4 through 14, inclusive.1196

And if you figure that out, that is actually 11 terms, because you are including 14.1206

So, starting with 4 and going up through 14, there are actually 11 terms, so n = 11.1209

It is really a11 = 15 I am asked to find the sum of these 11 terms.1215

And I can do that now, because I know that I have n = 11, divided by 2, and then I am going to get the first term1222

(that is 5), plus the last term, which is 25; so the sum is 11/2, times 5 + 25 (is 30).1228

Therefore, this cancels; I am going to get 11 times 15, and that is simply 165.1239

OK, so in sigma notation, this gives me a lot of information, because I saw that I knew the formula to find a particular term.1251

And I knew the n for the first term and the n for the last term.1260

So, I knew I could use this formula, because I could find the first and last terms.1264

So, I made n equal to 4 to find the first term, which is 5; I made n equal to 14, which is to help me find the last term, which is 25.1268

And then, I knew that, since it was going from 4 to 14, that n is equal to 11.1277

Once I had first term, last term, and n, it was just a matter of calculating the sum, which was 165.1282

That finishes up today's lesson on arithmetic series; thanks for visiting Educator.com!1290

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